Theorem nnssn0s 28740

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28734	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4075	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 3959	1 ⊢ ℕs ⊆ ℕ0s

This proof depends on syntax axioms:   ⊆ wss 3900   ∖ cdif 4048  {csn 4588   0_s c0s 28224  ℕ_0scn0s 28731  ℕ_scnns 28732

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-dif 4050  df-nns 28734

This theorem is used by:  nnssno  28741  nnn0s  28746  nnn0sd  28747  nnsgt0  28758